Solve $\sin x=0.5$ for $0^\circ\le x\le360^\circ$.

$\sin^{-1}0.5=30^\circ$. The sine graph is symmetrical about $90^\circ$, so the other solution is $180-30=150^\circ$. Answer: $30^\circ,\ 150^\circ$

Solve $\tan x=-1.5$ for $0^\circ\le x\le360^\circ$.

$\tan^{-1}(-1.5)=-56.3^\circ$, outside the range. tan has period $180^\circ$: add 180 and 360. $x=123.7^\circ$ or $303.7^\circ$. Answer: $123.7^\circ,\ 303.7^\circ$

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Edexcel IGCSE Maths (4MA1) · Unit 4: Graphs & Functions

Edexcel IGCSE 4MA1 Trigonometric Graphs: exam questions and answers, notes and worked examples

Updated October 2026 · By Pete Bromfield

The graphs of $y = \sin x$ and $y = \cos x$ are waves between $- 1$ and $1$ that repeat every $360^{\circ}$ ; $y = \tan x$ repeats every $180^{\circ}$ and has asymptotes at $90^{\circ}$ and $270^{\circ}$ , where it is undefined. Their symmetry gives you every solution of an equation such as $\sin x = 0.5$ in a range: your calculator gives one angle, and the graph shows where the others are.

Unit 4: Graphs & Functions
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Notes
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Worked examples
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FAQ

Trigonometric Graphs notes

The sine and cosine graphs

$y = \sin x$ starts at 0, peaks at $(90^{\circ}, 1)$ , crosses the axis at $180^{\circ}$ , reaches $- 1$ at $270^{\circ}$ and is back to 0 at $360^{\circ}$ . $y = \cos x$ is the same wave moved $90^{\circ}$ to the left: it starts at $(0^{\circ}, 1)$ and is $- 1$ at $180^{\circ}$ .

The tangent graph

$y = \tan x$ is 0 at $0^{\circ}$ , $180^{\circ}$ and $360^{\circ}$ , equals 1 at $45^{\circ}$ , and has vertical asymptotes at $x = 90^{\circ}$ and $x = 270^{\circ}$ . It repeats every $180^{\circ}$ and, unlike sine and cosine, takes every value.

Solving trig equations with the graphs

Find one angle with the inverse function, then use the graph. For sine, the second solution is $180^{\circ}$ minus the first: $\sin x = 0.5$ gives $30^{\circ}$ and $150^{\circ}$ . For cosine it is $360^{\circ}$ minus the first: $\cos x = 0.5$ gives $60^{\circ}$ and $300^{\circ}$ . For tan, add $180^{\circ}$ : $\tan x = 1$ gives $45^{\circ}$ and $225^{\circ}$ . The graph shows worked example 1.

Graph of y = sin x from 0° to 360° and the line y = 0.5, crossing at x = 30° and x = 150°, the two solutions of sin x = 0.5 — Worked example 1 drawn with our Grapher (x in degrees): the line y = 0.5 crosses the sine curve twice between 0° and 360°.

How many solutions?

The number of solutions is the number of times the horizontal line crosses the curve in the range. Because $- 1 \leq \sin x \leq 1$ , the equation $\sin x = 1.5$ has no solutions, while $\sin x = 1$ has exactly one between $0^{\circ}$ and $360^{\circ}$ , at $90^{\circ}$ .

Key formulas

Exact values Learn it: $\sin 30^{\circ} = \frac{1}{2}, \cos 60^{\circ} = \frac{1}{2}, \tan 45^{\circ} = 1, \sin 60^{\circ} = \frac{\sqrt{3}}{2}, \sin 45^{\circ} = \frac{\sqrt{2}}{2}$

Learn it = not printed on the 4MA1 Higher formulae sheet, so learn it by heart. The whole 4MA1 formula sheet (free PDF) → · 4MA1 formulas to learn

Trigonometric Graphs worked examples

Worked example 1: Solving with the sine graph

Solve $\sin x = 0.5$ for $0^{\circ} \leq x \leq 360^{\circ}$ .

Solution

$\sin^{- 1} 0.5 = 30^{\circ}$ .
The sine graph is symmetrical about $90^{\circ}$ , so the other solution is $180 - 30 = 150^{\circ}$ .

3 steps · 1 to fill in

Tip: Sketch the graph to see how many solutions there are.

Worked example 2: Solving tan x = k

Solve $\tan x = - 1.5$ for $0^{\circ} \leq x \leq 360^{\circ}$ .

Solution

$\tan^{- 1} (- 1.5) = - {56.3}^{\circ}$ , outside the range.
tan has period $180^{\circ}$ : add 180 and 360.
$x = {123.7}^{\circ}$ or ${303.7}^{\circ}$ .

4 steps

Tip: Solutions of tan x = k are always 180° apart.

Worked example 3: Solving tan x = k to 1 decimal place · 3 marks

Solve $\tan x = 2$ for $0^{\circ} \leq x \leq 360^{\circ}$ . Give your answers to 1 decimal place.

Mark scheme

M1 tan⁻¹ 2
A1 63.4°
B1 243.4° ft their first angle + 180

Answer: $x = {63.4}^{\circ}, {243.4}^{\circ}$

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Trigonometric Graphs exam-style questions

Try each question before you open the answer (one has its full worked solution open). The mark schemes use Edexcel-style marks: M for method, A for accuracy (after the method mark), B for an independent result.

Question 1 · easy · 1 mark

Write down the period of the graph of $y = \tan x$ .

Show the answer and mark scheme

B1 180°

Answer: 180°

Question 2 · easy · 2 marks

Write down the equations of the asymptotes of the graph of $y = \tan x$ for $0^{\circ} \leq x \leq 360^{\circ}$ .

Show the answer and mark scheme

B1 x = 90
B1 x = 270

Answer: $x = 90^{\circ}$ and $x = 270^{\circ}$

Question 3 · easy · 1 mark

Given that $\tan 45^{\circ} = 1$ , use the graph of $y = \tan x$ to write down the value of $\tan 225^{\circ}$ .

Show the answer and mark scheme

B1 1

Answer: 1

Question 4 · medium · 3 marks

Solve $\tan x = - 0.7$ for $0^{\circ} \leq x \leq 360^{\circ}$ . Give your answers to 1 decimal place.

Show the answer and mark scheme

M1 tan⁻¹(−0.7) = −35.0 (or 35.0 used)
A1 145.0°
A1 325.0°

Answer: $x = {145.0}^{\circ}, {325.0}^{\circ}$

Question 5 · medium · 2 marks

Given that $\tan 70^{\circ} \approx 2.747$ , write down (a) another angle between $0^{\circ}$ and $360^{\circ}$ whose tangent is 2.747, (b) an angle between $0^{\circ}$ and $360^{\circ}$ whose tangent is $- 2.747$ .

Full worked solution Free, no sign-in

B1 250°
B1 110° or 290°

Answer: (a) 250° (b) 110° (or 290°)

Every question here has a full worked solution. See them all →

Question 6 · hard · 2 marks

How many solutions does the equation $\tan x = 5$ have for $- 360^{\circ} \leq x \leq 360^{\circ}$ ? Give a reason.

Show the answer and mark scheme

M1 one solution found (78.7°) and period 180° used
A1 4

Answer: 4

Question 7 · hard · 3 marks

Solve $3 \tan x + 4 = 0$ for $0^{\circ} \leq x \leq 360^{\circ}$ . Give your answers to 1 decimal place.

Show the answer and mark scheme

M1 tan x = −4/3
A1 126.9°
A1 306.9°

Answer: $x = {126.9}^{\circ}, {306.9}^{\circ}$

Common trigonometric graphs mistakes

Stopping at one solution. Your calculator gives one angle. Use the graph's symmetry to find every solution in the range the question gives.
The sine rule for a cosine equation. The second solution of $\cos x = k$ is $360^{\circ}$ minus the first, not $180^{\circ}$ minus it: $\cos 120^{\circ} = - 0.5$ , not $0.5$ .
A first angle outside the range. $\tan^{- 1} (- 1.5) = - {56.3}^{\circ}$ is not between $0^{\circ}$ and $360^{\circ}$ ; add $180^{\circ}$ and $360^{\circ}$ to bring it into range.
Calculator in radians. Check the calculator is in degrees before using $\sin^{- 1}$ , $\cos^{- 1}$ or $\tan^{- 1}$ . An answer of 0.524 for $\sin x = 0.5$ is in radians.

Where students lose marks across 4MA1, from the examiner reports →

Trigonometric Graphs FAQ

What is the period of y = tan x?

180°. The tan graph repeats every 180°, so the solutions of tan x = k are 180° apart. The sine and cosine graphs repeat every 360°.

Are the trig graphs on the 4MA1 formula sheet?

No. Learn the shapes of y = sin x, y = cos x and y = tan x with their key points, and the exact values such as sin 30° = 0.5, which are not on the 4MA1 Higher formulae sheet.

Can I use a calculator for trigonometric graphs in 4MA1?

Yes. A calculator may be used in all four 4MA1 papers, so both Higher papers (1H and 2H) allow one. Most marks are for the method you write down, so show your working.

How long are the 4MA1 Higher papers?

Papers 1H and 2H are 2 hours each and 100 marks each, worth 50% each. Each tier has its own formulae sheet in the exam.

Next steps

Graphs, Sequences and Calculus revision notes · 4MA1 formula sheet · Taking Cambridge 0580 instead? Cambridge 0580 trigonometric graphs and equations questions

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Edexcel IGCSE 4MA1 Trigonometric Graphs: exam questions and answers, notes and worked examples

Try it first

Trigonometric Graphs notes

The sine and cosine graphs

The tangent graph

Solving trig equations with the graphs

How many solutions?

Key formulas

Trigonometric Graphs worked examples

Worked example 1: Solving with the sine graph

Solution

Worked example 2: Solving tan x = k

Solution

Worked example 3: Solving tan x = k to 1 decimal place · 3 marks

Mark scheme

Trigonometric Graphs exam-style questions

Question 1 · easy · 1 mark

Question 2 · easy · 2 marks

Question 3 · easy · 1 mark

Question 4 · medium · 3 marks

Question 5 · medium · 2 marks

Full worked solution Free, no sign-in

Question 6 · hard · 2 marks

Question 7 · hard · 3 marks

Common trigonometric graphs mistakes

Trigonometric Graphs FAQ

Next steps

Practise trigonometric graphs with your progress saved

More Unit 4 topics: Graphs & Functions